–what should the PE on the S&P 500 be if the main investment alternative for US investors, Treasury bonds, yield something around 4%?

On the first, over my 38+ year investment career stocks have gone mostly sideways when the Fed is raising short-term interest rates. The standard explanation for this, which I think is correct, is that while stocks can show rising earnings to counter the effect of better yields on newly-issued bonds, existing bonds have no defense.

Put a different way, the market’s PE multiple should contract as rates rise, but rising earnings counter at least part of that effect.

The second question, which is not about how we get there but what it looks like when we arrive, is the subject of this post.

in a 4% world

The arithmetic solution to the question is straightforward. Imagining that stocks are quasi-bonds, in the way traditional finance academics do, the equivalent of a bond coupon payment is the earnings yield. It’s the portion of a company’s profits that each share has a claim on ÷ the share price. For example, if a stock is trading at $50 a share and eps are $2, the earnings yield is $2/$50 = 4%. This is also 1/PE.

A complication: Ex dividends, corporate profits don’t get deposited into our bank accounts; they remain with management. So they’re somewhat different from an interest payment. If management is a skillful user of capital, that’s good. Otherwise…

If we take this proposed equivalence at face value, a 4% earnings yield and 4% T-bond annual interest payment should be more or less the same thing. In the ivory tower universe, stocks should trade at 25x earnings if T-bonds are yielding 4%. That’s almost exactly where the S&P 500 is trading now, based on trailing 12-months “as reported” earnings (meaning not factoring out one-time gains/losses). Why this measure? It’s the easiest to obtain.